Let $f(x)$ be any continuous function, then is it true that $$Z^{+}\left(\alpha f(x)+(x+\beta)f'(x)\right)\leq Z^{+}\left(f'(x)\right)$$

where $\alpha>1$ is a real number and $\beta$ is any positive integer. $Z^+$ denotes the number of positive zeros.

Note: If $f(x)$ is a polynomial then it's easy to see that the above inequality holds.

Any help or small hint will be really appreciated. Thanks.